On the Stabilizing Properties of Energy-Momentum Integrators and Coordinate Projections for Constrained Mechanical Systems

Energy Considerations for the Stabilization of Constrained Mechanical Systems with Velocity Projection

“…However, matrix P defined in (15) is not, in general, positive semidefinite, or even symmetric. As a consequence, the damping matrix D given by (16) will not be symmetric, and nothing can be said in general about its definiteness. This means that, following this procedure, it is not possible to bring the sign of the energy balance ΔE p forward at each time step.…”

“…Nevertheless, exact conservation of energy (or unconditional energy dissipation) has revealed itself to be extremely useful in the design of robust integration schemes, with excellent stability in the nonlinear case ( [24] and references therein) and applied to the dynamics of multibody systems [14,18,15,16].…”

“…where the symmetric matrix D s can be expressed, after some algebraic manipulations using definition (16), as:…”

Energy considerations for the stabilization of constrained mechanical systems with velocity projection

“…This means that, in order to analyze the properties of the quadratic form given by (16), it is possible to work just with the symmetric part of matrix D. Denoting the symmetric and skew-symmetric parts of the original matrix by superscripts s and h, respectively, this result may be expressed as:…”

A Conservative Augmented Lagrangian Algorithm for the Dynamics of Constrained Mechanical Systems

“…Exact discrete energy conservation does not guarantee unconditional stability for the non linear case, but definitely enhance it, as it has been widely reported in the literature (Stuart and Humphries, 1996;García Orden and Dopico Dopico, 2006). This is the main motivation behind this work, that aims to extend a previously developed conserving penalty formulation Goicolea and García Orden, 2000;García Orden and Goicolea, 2001), which showed excellent stability properties, to the augmented Lagrangian formulation, which is the main contribution of this paper and it is developed in Section 3.2.…”